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Stability Spectrum
In model theory, a branch of mathematical logic, a complete theory, complete first-order theory ''T'' is called stable in λ (an infinite cardinal number), if the Type (model theory)#Stone spaces, Stone space of every structure (mathematical logic), model of ''T'' of size ≤ λ has itself size ≤ λ. ''T'' is called a stable theory if there is no upper bound for the cardinals κ such that ''T'' is stable in κ. The stability spectrum of ''T'' is the class of all cardinals κ such that ''T'' is stable in κ. For countable theories there are only four possible stability spectra. The corresponding dividing line (model theory), dividing lines are those for totally transcendental theory, total transcendentality, superstable theory, superstability and stable theory, stability. This result is due to Saharon Shelah, who also defined stability and superstability. The stability spectrum theorem for countable theories Theorem. Every countable complete first-order theory ''T'' falls into o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Totally Transcendental Theory
In the mathematical field of model theory, a Theory (mathematical logic), theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory (model theory), classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Superstable Theory
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common goa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stable Theory
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Spectrum Of A Theory
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory ''T'' in a language we write ''I''(''T'', ''κ'') for the number of models of ''T'' (up to isomorphism) of cardinality ''κ''. The spectrum problem is to describe the possible behaviors of ''I''(''T'', ''κ'') as a function of ''κ''. It has been almost completely solved for the case of a countable theory ''T''. Early results In this section ''T'' is a countable complete theory and ''κ'' is a cardinal. The Löwenheim–Skolem theorem shows that if ''I''(''T'',''κ'') is nonzero for one infinite cardinal then it is nonzero for all of them. Morley's categoricity theorem was the first main step in solving the spectrum problem: it states that if ''I''(''T'',''κ'') is 1 for some uncountable ''κ'' then it is 1 for all uncountable ''κ''. Robert Vaught showed that ''I''(''T'',ℵ0) can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Group Of Finite Morley Rank
In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below). Examples *A group of finite Morley rank is an abstract group ''G'' such that the formula ''x'' = ''x'' has finite Morley rank for the model ''G''. It follows from the definition that the theory of a group of finite Morley rank is ω-stable; therefore groups of finite Morley rank are stable groups. Groups of finite Morley rank behave in certain ways like finite-dimensional objects. The striking similarities between groups of finite Morley rank and finite groups are an object of active research. *All finite groups have finite Morley rank, in fact rank 0. *Algebraic groups over algebraically closed fields have finite Morley rank, equal to their dimension as algebraic sets. * showed that free groups, and more generally torsion-free hyperbolic groups, are stable. Free groups on more than one genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Morley's Categoricity Theorem
In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers \mathbb. In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Morley Rank
In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model theory, model of a theory (logic), theory, generalizing the notion of dimension in algebraic geometry. Definition Fix a theory ''T'' with a model ''M''. The Morley rank of a formula ''φ'' defining a definable set, definable (with parameters) subset ''S'' of ''M'' is an ordinal number, ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least ''α'' for some ordinal ''α''. *The Morley rank is at least 0 if ''S'' is non-empty. *For ''α'' a successor ordinal, the Morley rank is at least ''α'' if in some elementary extension ''N'' of ''M'', the set ''S'' has countably infinitely many disjoint definable subsets ''Si'', each of rank at least ''α'' − 1. *For ''α'' a non-zero limit ordinal, the Morley rank is at least ''α'' if it is at least ''β'' for all ''β'' less than ''α''. The Morley ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Saharon Shelah
Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Hebrew poet and Canaanist political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Dividing Line (model Theory) , an album by the progressive metal band Threshold released in 2022
{{DEFAULTSORT:Dividing Line, The ...
The Dividing Line may refer to: * ''The Dividing Line'' (Youth Brigade album) * ''The Dividing Line'' (SSS album) * Alternative title for the 1950 American film ''The Lawless'' * "The Dividing Line" (song), by Genesis on the 1997 album ''Calling All Stations'' See also * Dividing Lines ''Dividing Lines'' is the twelfth studio album by progressive metal band Threshold, released on 18 November 2022. Track listing Personnel Threshold * Glynn Morgan – vocals * Karl Groom – guitars, production, mixing, mastering * Ric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures of first-order theories with no relation symbols. Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a '' semantic model'' when one discusses the notion in the more general setting of mathematical models. Log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |